Table Of ContentNORTH-HOLLAND
MATHEMATICS STUD I ES 28
Notas de Matemhtica (64)
Editor: Leopoldo Nachbin
Universidade Federal do Rio de Janeiro
and University of Rochester
Saks Spaces
and Applications to Functional Analysis
JAMES BELL COOPER
Johannes-Kepler Universitat. Linz, Austria
1978
-
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD
@ North-Holland Publishing Company - 1978
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Library of Congress Cataloging in Publication Data
Cooper, James Bell.
Saks spaces and applications to functional analysis.
(Notas de matemhtica ; 64) (North-Holland mathe-
matics studies)
Bibliography: p.
Includes index.
1. Saks spaces. 2. Functional analysis.
I. Title. 11. Series.
@l.N86 no. 64 [@I3221 510'.8s [515'.73] 78-1921
ISBN o-wA-~~~oo-~
PRINTED INT HE NETHERLANDS
PREFACE
This monograph is concerned with two streams in functional
-
analysis the theories of mixed topologies and of strict topo-
logies. The first deals with mathematical objects consisting
of a vector space together with a norm and a locally convex
topology which are in some sense compatible. Their theory can
be regarded as a generalisation of that of Banach spaces,
complementary to the theory of locally convex spaces. Although
closely related to locally convex spaces, the theory of mixed
spaces (or Saks spaces as we propose to call them) has its own
flavour and requires its own special techniques. The second
theory is concerned with a number of special topologies on spaces
of functions and operators which possess the common property that
they are substitutes for natural norm topologies which are not
suitable for certain applications. The best known example of
the latter is the strict topology on the space of bounded, con-
tinuous, complex-valued functions on a locally compact space,
which was introduced by Buck. The connection between these two
topics is provided by the fact that the important strict topolo-
gies can be regarded in a natural way as Saks spaces and this
fact allows a simple and unified approach to their theory.
The author feels that the theory of Saks spaces is sufficiently
well-developed and useful to justify an attempt at a first syn-
thesis of the theory and its applications. The present monograph
is the consequence of this conviction.
V
vi PREFACE
The book is divided into five chapters, devoted successively
to the general theory of Saks spaces and to important spaces
of functions or operators with strict topologies (spaces of
bounded cont nuous functions, bounded measurable functions,
operators on Hilbert spaces and bounded holomorphic functions
.
respectively An appendix contains a category-theoretical
approach to Saks spaces. Each chapter is divided into sections
and Propositions, Definitions etc. are numbered accordingly.
Hence a reference to 1.1.1 is to the first definition (for
example) in 9 1 of Chapter I. Within Chapter I this would be
abbreviated to 1.1. At the end of the book, indexes of notations
and terms have been added.
An attempt has been made to make the first five chapters as
self-contained as possible so that they will be useful and ac-
cessible to non-specialists. However, some results have been
given without proof, either because they are standard or because
they seemed to the author to provide useful insight on the
subject but the proofs were unsuitable for inclusion. Some of
the latter are relegated to Remarks. In the first Chapter, only
familiarity with the basic concepts of Banach spaces and locally
-
convex spaces is assumed. In Chapter I1 V the principle of
-
GILLMAN and JERISON has been applied namely to make them com-
prehensible to a reader who understands the words in their re-
spective titles. The monograph should therefore be suitable say
for a graduate course or seminar. However, by providing each
Chapter with notes (brief historical remarks and references to
PREFACE vii
related research articles) and a list of references, an attempt
has been made to produce a work which could also be useful to
research workers in this and related fields. In addition to
those references which we have used directly in the preparation
of this book, we have tried to include a complete bibliography
of papers which relate to Saks spaces and their applications,
including some where the connection may seem rather remote.
As a warning to the reader, we mention that all topological
spaces (and hence also locally convex spaces, uniform spaces etc.)
are tacitly assumed to be Hausdorff.
It goes without saying that the author gratefully acknowledges
the contributions to this book of all those mathematicians whose
published works, preprints, correspondence or conversation with
the author have been used in its preparation. Special thanks
are due to Frxulein G. Jahn who made a beautiful job of typing
a manuscript in a foreign language under rather trying circum-
stances.
J.B. Cooper
Edramsberg, October, 1977
-
CHAPTER I MIXED TOPOLOGIES
Introduction: The objects of study in this chapter are Banach
spaces with a supplementary structure in the form of an additio-
nal locally convex topology. The motivation lies in the inter-
play between certain mathematical objects (topological spaces,
measure spaces etc.) and suitable spaces of (complex-valued)
functions on them. These often have a natural Banach space
structure. However, by passing over from the original spaces
I
to the associated Banach spaces, one frequently loses crucial
information on the underlying space. A good example (which
will be the subject of our most important application of mixed
topologies) is the Banach space Cm(S) of bounded, continuous,
compl x-valued functions on a locally compact space S where
it is impossible to recover S from the Banach space structure
of cm S) (in contrast to the case of compact spaces S ) .
As we shall see in Chapter 11, this situation can be saved by
enriching the structure of Cm(S) with the topology ' I ~of uniform
convergence on the compact subsets of S. The class of spaces
that we consider can be regarded as a generalisation of the
class of Banach spaces (we can "enrich" a Banach space in a
trivial way, namely by adding its own topology). In fact these
spaces can be regarded as projective limits of certain spectra
of Banach spaces with contractive linking mappings (just as
one can regard (complete) locally convex spaces as projective
limits of arbitrary spectra of Banach spaces) and we shall
lay particular emphasis on this fact for two reasons: for
1
2 I. MIXED TOPOLOGIES
purely technical reasons and secondly because, in applications
to function spaces, we shall constantly use the fact that our
function spaces are constructed out of simpler blocks which
correspond exactly to the members of a representing spectrum
of Banach spaces. As an example, dual to the fact that one
can consider a locally compact space as being built up from
its compact subspaces, we find that one can construct the
space Cm (S) from the spectrum defined by the spaces {C (K)1 as
K runs through these subsets.
One of our main tools in the study of our enriched Banach
-
spaces will be a natural locally convex topology the mixed
topology of the title of this chapter. It turns out that
this can be regarded as a generalised type of inductive limit.
The latter were systematically studied by GARLING in his disser-
tation. For this reason, we begin with a treatment of this
theory in the generality suitable to our purposes.
For the convenience of the reader, we now give a brief summary
of Chapter I. In the first section, we give a basic treatment
of generalised inductive limits. Essentially, we consider a
vector space with two locally convex topologies which satisfy
suitable compatibility conditions. We then introduce in a
natural way a "mixed topology" and this section is devoted
to relating its properties to those of the original topologies
However, a closer examination of the definitions and results
shows that, for one of the topologies, only the bounded sets
INTRODUCTION 3
are relevant. We have taken the consequences of this obser-
vation by replacing this topology by a "bornology", that
is a suitable collection of sets which satisfy properties
which one would expect of a family of bounded sets. We really
only use the language (and not the theory) of bornologies
and introduce explicitly all the terms that we use. In
section 2, we give a list of examples of spaces with mixed
topologies. Some of these will be studied in detail (and
in more generality) in the following chapters. Others are
introduced to supply counter-examples. All are used to
illustrate the ideas of the first section. In section three,
we define the class of enriched Banach spaces mentioned, restate
the results of sections 1 in the form that we shall require
them for applications and describe the usual methods for con-
structing new spaces (subspaces, products, tensor products etc.).
It is perhaps not inappropriate to mention here that one of
the main reasons for our emphasis on spaces with two structures
(a norm and a locally convex topology) rather than on locally
convex spaces of a rather curious type is the fact that it is
important that these constructions be so carried out that
this double structure is preserved and not in the sense of
locally convex spaces. The fourth section is devoted to
attempts to extend the classical results on Banach spaces to
enriched Banach spaces (e.g. Banach-Steinhaus theorem, closed
graph theorem). The results obtained are perhaps rather un-
satisfactory since they involve special hypotheses but, as we
shall see later, they can often be applied to important function
4 I. MIXED TOPOLOGIES
spaces. In any case, there are simple counter-examples bhich
show that such results cannot hbld without rather special
restrictions.
I. 1. BASIC THEORY
As announced in the Introduction to this chapter, it is con-
venient for us to use the language of bornologies. We begin
with their definition:
1.1. Definition: Let E be a vector space. A ball in E is an
absolutely convex subset of E which does not contain a non-
trivial subspace. If B is a ball in El we write EB for the
03
linear span U nB of B in E. Then
n=l
11 /IB : x + inf (X > 0 : x E XB)
llB)
is a normon E. If (EB,II is a Banach space, B is a
Banach ball.
Note that any absolutely convex, bounded subset of a locally
convex space is a ball. The following Lemma qives a sufficient
(but not necessary) condition for it to be a Banach ball.
1.2. Lemma: Let B be a bounded ball in a locally convex space
(E,T). Then if B is sequentially.complete for T (and in particu-
lar if it is T-complete), B is a Banach ball.
I. 1 BASIC THEORY 5
Proof: Let (x,) be a Cauchy sequence in (EBl 11 ]IB). Then,
since B is bounded, (x,) is T-Cauchy. Hence there is an x E B
so that xn -+ x for T. We show that IIxn - xllB- 0.
If E > 0, there is an N EN so that (xm - xn) belongs to EB
for m,n 2 N. Since B (and so also EB) is sequentially complete
and so sequentially closed, we can take the limit over n to
deduce that xm - x belongs to EB for m 2 N.
1.3. Definition: If E is a vector space, a (convex) bornology
on E is a family B of balls in E so that
(a) E = U B;
(b) if B E 8, X > 0, then XB E 8;
(c) €3 is directed on the right by inclusion (i.e. if
B,C E B then there exists D E B with B u C 2 D);
.
(d) if B E B and C is a ball contained in B, then C E B
A subset B of E is B-bounded if it is contained in some ball
in 8.
A basis for B is a subfamily B1 of B so that each B E B is
a subset of some B1 E B1.
(E,B) is complete if B has a basis consisting of Banach balls.
B is of countable type if B has a countable basis.
If (E,T) is a locally convex space, then B,, the family of
all T-bounded, absolutely convex subsets of E is a bornology
-
on E the von Neumann bornology. In many of our applications B
.
will be the von Neumann bornology of a normed space (Ell[ 11 )
This is of countable type (the family {nB where B
II II II
I b E
is the unit ball of E is a basis).
